1. Let (X, Y,Z) be a point selected at random in the unit sphere



(x, y, z) : x2 + y2 + z2 ≤ 1

;

that is, the sphere of radius 1 centered at the origin. [Note that the volume of a sphere with radius R is (4/3)πR3.]

(a) Find

f, the joint probability density function of X, Y, and Z.

(b) Find the joint probability density function marginalized over X and Y .

(c) Find the joint probability density function marginalized over Z.

Hint: To find fZ(z), convert the Cartesian coordinates (x, y) to polar coordinates

(r, θ).